Question

U 10 I LI IU I LILIUI IUL 2. (Bases for column and row spaces) Suppose A is a 3 x 3 matrix with pivots in exactly the first and second columns. Suppose B is the unique reduced echelon form of A. (a) We know that the first and second columns of A form a basis for Col A. Is it necessarily true that the first and second columns of B form a basis for Col A? If so, explain why. If not, provide a counterexample. (b) We know that the first and second rows of B form a basis for Row A. Is it necessarily true that the first and second rows of A form a basis for Row A? If so, explain why. If not, provide a counterexample. 3. (The Rank Theorem) (a) Suppose a nonhomogeneous system of five linear equations in seven unknowns is solvable and the general solution has two free variables. Is it possible to change constants on the right-hand side to make the system inconsistant? Use the Rank Theorem to explain your answer. (b) Suppose a nonhomogeneous system of seven linear equations in five unknowns is solvable and the general solution has two free variables. Is it possible to change constants on the right-hand side to make the system inconsistant? Use the Rank Theorem to explain your answer.

Answer 1

Šl 66 - 9. A be a 3x3 matrix with pivots in first and Third Glumn GIA = span { Ist Glumn, ad Glumn of A3 then need And GIA = spanq Ist and and Column of B3 not possible where Bů Reduced echlorn from af A Lot Arrio 07 A is a then reduced echlon form af so, G1A= { span (5),(6)3 (cuius is bau's of GIA) + span{ (3), (4) 3 Also, row (Ale spank Ist, and row of A3 ie. {Ist, and row } i i a sasis of Row (A) Ist and 3rd row of Breed not to be from a 1 far Row (AI basis 1. rió 02 exa, hot A Too then B=A Row (A la spanſ (1010), (010133# spanſ (lov), (0.003 1. So, {(1,010), 101010) } doesn't fooms a sasis for lowlan